Single World Intervention Graphs (SWIGs): Unifying the - - PowerPoint PPT Presentation
Single World Intervention Graphs (SWIGs): Unifying the Counterfactual and Graphical Approaches to Causality Thomas Richardson Department of Statistics University of Washington Joint work with James Robins (Harvard School of Public Health)
Unifying the Counterfactual and Graphical Approaches to Causality
Thomas Richardson Department of Statistics University of Washington
Joint work with James Robins (Harvard School of Public Health)
Therme Vals Causal Workshop 5 Aug 2013
Brief review of counterfactuals A new unification of graphs and counterfactuals via node-splitting
◮ Factorization and Modularity ◮ Contrast with Twin Network approach ◮ Some Examples and Extensions ◮ Sequentially Randomized Experiments / Time Dependent
Confounding
◮ Dynamic Regimes
Experimental Testability and Independence of Errors in NPSEMs
Thomas Richardson Therme Vals Workshop Slide 1
Thomas Richardson Therme Vals Workshop Slide 2
Hume (1748) An Enquiry Concerning Human Understanding:
We may define a cause to be an object followed by another, and where all the objects, similar to the first, are followed by objects similar to the second, . . . . . . where, if the first object had not been the second never had existed.
Thomas Richardson Therme Vals Workshop Slide 3
Jerzy Neyman (1923):
To compare v varieties [on m plots] we will consider numbers: U11
. . .
U1m
. . . . . .
Uv1
. . .
Uvm
Here Uij is the crop yield that would be observed if variety i were planted in plot j. Physical constraints only allow one variety to be planted in a given plot in any given growning season.
Popularized by Rubin (1974); sometimes called the ‘Rubin causal model’.
Thomas Richardson Therme Vals Workshop Slide 4
For binary treatment X and response Y, we define two potential
Y(x = 0): the value of Y that would be observed for a given
unit if assigned X = 0;
Y(x = 1): the value of Y that would be observed for a given
unit if assigned X = 1; WIll also write these as Y(x0) and Y(x1). Implicit here is the assumption that these outcomes are well-defined. Specifically:
◮ Only one version of treatment X = x ◮ No interference between units (SUTVA). Thomas Richardson Therme Vals Workshop Slide 5
Unit Potential Outcomes Observed
Y(x = 0) Y(x = 1) X Y
1 1 2 1 3 4 1 1 5 1
Thomas Richardson Therme Vals Workshop Slide 6
In the simplest case where Y is a binary outcome we have the following 4 types:
Y(x0) Y(x1)
Name Never Recover 1 Helped 1 Hurt 1 1 Always Recover
Thomas Richardson Therme Vals Workshop Slide 7
Unit Potential Outcomes Observed
Y(x = 0) Y(x = 1) X Y
1 1 1 2 1 3 1 4 1 1 1 5 1
Thomas Richardson Therme Vals Workshop Slide 8
Unit Potential Outcomes Observed
Y(x = 0) Y(x = 1) X Y
1 1 1 1 2 1 3 1 4 1 1 1 1 5 1 1
Thomas Richardson Therme Vals Workshop Slide 9
Unit Potential Outcomes Observed
Y(x = 0) Y(x = 1) X Y
1 ? 1 1 1 2 ? 3 ? 1 4 ? 1 1 1 5 1 ? 1
Thomas Richardson Therme Vals Workshop Slide 10
ACE(X → Y)
≡ E[Y(x1) − Y(x0)] = p(Helped) − p(Hurt) ∈ [−1, 1]
Thus ACE(X → Y) is the difference in % recovery if everyone treated (X = 1) vs. if noone treated (X = 0).
Thomas Richardson Therme Vals Workshop Slide 11
If X is assigned randomly then
X ⊥ ⊥ Y(x0)
and
X ⊥ ⊥ Y(x1)
(1) hence
E[Y(x1) − Y(x0)] = E[Y(x1)] − E[Y(x0)] = E[Y(x1) | X = 1] − E[Y(x0) | X = 0] = E[Y | X = 1] − E[Y | X = 0].
Thus if (1) holds then ACE(X → Y) is identified from P(X, Y).
Thomas Richardson Therme Vals Workshop Slide 12
Suppose that we do not know that X ⊥
⊥ Y(x0) and X ⊥ ⊥ Y(x1).
What can be inferred?
X = 0 X = 1
Placebo Drug
Y = 0
200 600
Y = 1
800 400 What is: The largest number of people who could be Helped? 400 + 200 The smallest number of people who could be Hurt? 0
⇒ Max value of ACE: (200 + 400)/2000 − 0 = 0.3
Similar logic:
⇒ Min value of ACE: 0 − (600 + 800)/2000 = −0.7
Thomas Richardson Therme Vals Workshop Slide 13
Suppose that we do not know that X ⊥
⊥ Y(x0) and X ⊥ ⊥ Y(x1).
General case:
−(P(x=0, y=1) + P(x=1, y=0)) ACE(X → Y) ACE(X → Y) P(x=0, y=0) + P(x=1, y=1) ⇒ Bounds will always cross zero. ⇒ X ⊥ ⊥ Y(x0) and X ⊥ ⊥ Y(x1) essential for drawing non-trivial
causal inferences.
Thomas Richardson Therme Vals Workshop Slide 14
In our observed data, for each unit one outcome will be ‘actual’; the others will be ‘counterfactual’. The potential outcome framework allows Causation to be ‘reduced’ to Missing Data
⇒ Conceptual progress!
The ACE is identified if X ⊥
⊥ Y(xi) for all values xi.
Randomization of treatment assignment implies X ⊥
⊥ Y(xi).
Ideas are central to Fisher’s Exact Test; also many parts of experimental design. The framework is the basis of many practical causal data analyses published in Biostatistics, Econometrics and Epidemiology.
Thomas Richardson Therme Vals Workshop Slide 15
Potential outcomes can be seen as a different notation for Non-Parametric Structural Equation Models (NPSEMs): Example: X → Y. NPSEM formulation:
Y = f(X, ǫY)
Potential outcome formulation:
Y(x) = f(x, ǫY)
Two important caveats: NPSEMs typically assume all variables are seen as being subject to well-defined interventions (not so with potential
Pearl associates NPSEMs with Independent Errors (NPSEM-IEs) with DAGs (more on this later).
Thomas Richardson Therme Vals Workshop Slide 16
Expressions in terms of ‘do’ can be expressed in terms of counterfactuals:
P(Y(x) = y) ≡ P(Y = y | do(X = x))
but counterfactual notation is more general. Ex. Distribution of
(X = 1) had counter-to-fact they not received treatment:
P(Y(x = 0) = y | X = 1)
If treatment is randomized, so X ⊥
⊥ Y(x = 0) then this equals P(Y(x = 0) = y), but in an observational study these may be
different.
Thomas Richardson Therme Vals Workshop Slide 17
Thomas Richardson Therme Vals Workshop Slide 18
X Y
No Confounding
X H Y
Confounding Unobserved
Graph intended to represent direct causal relations. Convention that confounding variables (e.g. H) are always included
Approach originates in the path diagrams introduced by Sewall Wright in the 1920s. If X → Y then X is said to be a parent of Y; Y is child of X.
Thomas Richardson Therme Vals Workshop Slide 19
X Y P(X, Y) = P(X)P(Y | X)
No Confounding
X Y P(X, Y) = P(Y)P(X | Y)
No Confounding Neither factorization places any restriction on P(X, Y).
Thomas Richardson Therme Vals Workshop Slide 20
X Y X ⊥ ⊥ Y(x0) & X ⊥ ⊥ Y(x1) X H Y X ⊥ ⊥ Y(x0) & X ⊥ ⊥ Y(x1)
Unobserved Elephant in the room: The variables Y(x0) and Y(x1) do not appear on these graphs!!
Thomas Richardson Therme Vals Workshop Slide 21
X Y P(X= ˜ x, Y = ˜ y) = P(X= ˜ x)P(Y = ˜ y | X= ˜ x)
X x = 0 Y(x = 0)
Can now ‘read’ the independence: X ⊥
⊥ Y(x=0).
Also associate a new factorization:
P (X= ˜ x, Y(x=0)= ˜ y) = P(X= ˜ x)P (Y(x=0)= ˜ y)
where:
P (Y(x=0)= ˜ y) = P(Y = ˜ y | X=0).
This last equation links a term in the original factorization to the new factorization. We term this the ‘modularity assumption’.
Thomas Richardson Therme Vals Workshop Slide 22
X Y P(X= ˜ x, Y = ˜ y) = P(X= ˜ x)P(Y = ˜ y | X= ˜ x)
X x = 1 Y(x = 1)
Can now ‘read’ the independence: X ⊥
⊥ Y(x=1).
Also associate a new factorization:
P (X= ˜ x, Y(x=1)= ˜ y) = P(X= ˜ x)P (Y(x=1)= ˜ y)
where:
P (Y(x=1)=y) = P(Y =y | X=1).
Thomas Richardson Therme Vals Workshop Slide 23
Crucial point: Y(x=0) and Y(x=1) are never on the same graph. Although we have:
X ⊥ ⊥ Y(x=0)
and
X ⊥ ⊥ Y(x=1)
we do not have
X ⊥ ⊥ Y(x=0), Y(x=1)
Had we tried to construct a single graph containing both Y(x=0) and Y(x=1) this would have been impossible. (Why?)
⇒ Single-World Intervention Graphs (SWIGs).
Thomas Richardson Therme Vals Workshop Slide 24
X Y G
X x Y(x) G(x)
Represent both graphs via a template: Formally this is a ‘graph valued function’: Takes as input a specific value x∗ Returns as output a SWIG G(x∗). Each instantiation of the template is a SWIG G(x∗) that represents a different margin: P(X, Y(x∗)) with red nodes x∗ becoming constants.
Thomas Richardson Therme Vals Workshop Slide 25
(Robins, VanderWeele, Richardson 2007)
Q: How could we identify whether someone would choose to take treatment, i.e. have X = 1, and at the same time find out what happens to such a person if they don’t take treatment Y(x = 0)? A: Consider an experiment in which, whenever a patient is
by administering a safe ‘emetic’ that causes the pill to be regurgitated before any drug can enter the bloodstream. Since we assume the emetic has no side effects, the patient’s recorded outcome is then Y(x = 0).
Thomas Richardson Therme Vals Workshop Slide 26
X0 Z H X1 Y
Query: does this causal graph imply?
Y(x0, x1) ⊥ ⊥ X1(x0) | Z(x0), X0,
Thomas Richardson Therme Vals Workshop Slide 27
X0 Z H X1 Y X0 x0 Z(x0) H X1(x0) x1 Y(x0, x1)
Query does this graph imply:
Y(x0, x1) ⊥ ⊥ X1(x0) | Z(x0), X0
? Answer: Yes – applying d-separation to the SWIG on the right we see that there is no d-connecting path from Y(x0, x1) given Z(x0). More on this shortly...
Thomas Richardson Therme Vals Workshop Slide 28
Given a graph G, a subset of vertices A = {A1, . . . , Ak} to be intervened
(1) (Node splitting): For every A ∈ A split the node into a random
node A and a fixed node a:
A · · · · · ·
A a
Splitting: Schematic Illustrating the Splitting of Node A The random half inherits all edges directed into A in G; The fixed half inherits all edges directed out of A in G.
Thomas Richardson Therme Vals Workshop Slide 29
(2) Relabel descendants of fixed nodes: a
A B C D F E X T Y Z
· · · · · · · · · · · · a
A(. . .) B(a, . . .) C(a, . . .) D(a, . . .) F(a, . . .) E(a, . . .) X(. . .) T(. . .) Y(. . .) Z(. . .)
· · · · · · · · · · · ·
Thomas Richardson Therme Vals Workshop Slide 30
A Single World Intervention Graph (SWIG) G(a∗) is obtained from the Template G(a) by simply substituting specific values a∗ for the variables a in G(a); For example, we replace G(x) with G(x=0). Changing the value of a fixed variable corresponds to constructing a new graph and considering a different population, e.g. P(X, Y(x=0)) vs. P(X, Y(x=1)) It is only the instantiated graph G(˜
x) that represents P(V(˜ x)),
not the template G(x).
Thomas Richardson Therme Vals Workshop Slide 31
Factorization: P(V(˜
a)) over the counterfactual variables in G(˜ a)
factorizes with respect to G(˜
a) (ignoring fixed nodes): P (V(˜ a)) =
a)∈V(˜ a)
P
a)
a)(Y(˜
a)) \ ˜ a
Modularity: P(V(˜
a)) and P(V) are linked as follows: P
a)=y
a)(Y(˜
a)) \ ˜ a
apaG(Y)∩A
So the conditional density associated with Y(˜ aY) in G(˜ a) is just the conditional density associated with Y in G after substituting ˜ ai for any Ai ∈ A that is a parent of Y.
Thomas Richardson Therme Vals Workshop Slide 32
Counterfactual conditional independence relations may be
adding fixed nodes to the conditioning set: Given disjoint subsets B(˜
a), C(˜ a) and D(˜ a) of random vertices
(where D(˜
a) may be empty),
if B(˜
a) is d-separated from C(˜ a) given D(˜ a) ∪ ˜ a in G(˜ a)
(2) then
B(˜ a) ⊥ ⊥ C(˜ a) | D(˜ a) [P(V(˜ a))].
In words, if in G(˜ a) two subsets B(˜ a) and C(˜ a) of random nodes are d-separated by D(˜ a) in conjunction with the fixed nodes ˜ a, then B(˜ a) and C(˜ a) are conditionally independent given D(˜ a) in the associated distribution P(V(˜ a)).
Thomas Richardson Therme Vals Workshop Slide 33
intuitive since these are fixed constants in the SWIG Since vertices in ˜
a have no parents, no new paths d-connect
due to also conditioning on ˜
a. ⇒ If a d-separation holds in G(˜ a) without conditioning on the
fixed nodes, then it will continue to hold if we also condition on fixed nodes. An alternative is simply to restrict attention to paths that do not contain fixed vertices, e.g. remove fixed nodes from the graph before checking d-separation.
Thomas Richardson Therme Vals Workshop Slide 34
Intervention on Z alone.
X Y Z X(˜ z) Z
˜
z Y(˜ z)
factorization:
P(Z, X(˜ z), Y(˜ z)) = P(Z)P(X(˜ z))P(Y(˜ z) | X(˜ z))
modularity:
P(X(˜ z)=x) = P(X=x | Z= ˜ z), P(Y(˜ z)=y | X(˜ z)=x) = P(Y =y | X=x, Z= ˜ z).
d-separation gives:
Z ⊥ ⊥ X(˜ z), Y(˜ z).
Thomas Richardson Therme Vals Workshop Slide 35
Intervention on Z and X:
X Y Z X(˜ z)
˜
x Z
˜
z Y(˜ x, ˜ z)
factorization:
P(Z, X(˜ z), Y(˜ x, ˜ z)) = P(Z)P(X(˜ z))P(Y(˜ x, ˜ z))
modularity:
P(X(˜ z)=x) = P(X=x | Z= ˜ z), P(Y(˜ x, ˜ z)=y) = P(Y =y | X= ˜ x, Z= ˜ z).
d-separation gives:
Z ⊥ ⊥ X(˜ z) ⊥ ⊥ Y(˜ x, ˜ z)
Thomas Richardson Therme Vals Workshop Slide 36
X Y Z X(˜ z)
˜
x Z
˜
z Y(˜ x)
factorization:
P(Z, X(˜ z), Y(˜ x)) = P(Z)P(X(˜ z))P(Y(˜ x))
modularity:
P(X(˜ z)=x) = P(X=x | Z= ˜ z), P(Y(˜ x)=y) = P(Y =y | X= ˜ x).
d-separation gives:
Z ⊥ ⊥ X(˜ z) ⊥ ⊥ Y(˜ x)
Thomas Richardson Therme Vals Workshop Slide 37
X0 Z H X1 Y X0 x0 Z(x0) H X1(x0) x1 Y(x0, x1)
Pearl (2009), Ex. 11.3.3, claims the causal DAG above does not imply: Y(x0, x1) ⊥ ⊥ X1 | Z, X0 = x0. (3) The SWIG shows that (3) does hold; Pearl is incorrect. Specifically, we see from the SWIG: Y(x0, x1) ⊥ ⊥ X1(x0) | Z(x0), X0, (4) ⇒ Y(x0, x1) ⊥ ⊥ X1(x0) | Z(x0), X0 = x0. (5) This last condition is then equivalent to (3) via consistency. (Pearl infers a claim of Robins is false since if true then (3) would hold).
Thomas Richardson Therme Vals Workshop Slide 38
X0 Z H X1 Y X0 Z H X1 Y x0 Z(x0, x1) H(x0, x1) x1 Y(x0, x1) UZ UH UY
The twin network fails to reveal that Y(x0, x1) ⊥ ⊥ X1 | Z, X0 = x0. This ‘extra’ independence holds in spite of d-connection because (by consistency) when X0 = x0, then Z = Z(x0) = Z(x0, x1). Note that Y(x0, x1) ⊥ ⊥ X1 | Z, X0 = x0. Shpitser & Pearl (2008) introduce a pre-processing step to address this.
Thomas Richardson Therme Vals Workshop Slide 39
X x Y(x) H
Here we can read directly from the template that X ⊥
⊥ Y(x) since
there is a path:
X ← H → Y(x).
Thomas Richardson Therme Vals Workshop Slide 40
X Y L X x Y(x) L
Here we can read directly from the template that
X ⊥ ⊥ Y(x) | L.
It follows that:
P(Y(˜ x)=y) =
P(Y =y | L=l, X= ˜ x)P(L=l).
(6)
Thomas Richardson Therme Vals Workshop Slide 41
X Y L X x Y(x) L X Y L
This ‘explains’ why L is sufficient to control confounding under the null (where X has no effect on Y) but not under the alternative.
Thomas Richardson Therme Vals Workshop Slide 42
X Y L X x Y(x) L X ⊥ ⊥ Y(x) | L.
Proof of identification:
P[Y(˜ x) = y] =
P[Y(˜ x) = y | L = l]P(L = l) =
P[Y(˜ x) = y | L = l, X = ˜ x]P(L = l) indep =
P[Y = y | L = l, X = ˜ x]P(L = l) modularity
Thomas Richardson Therme Vals Workshop Slide 43
X Y L H
(a-i)
X x Y(x) L H
(a-ii) Here we can read directly from the template that
X ⊥ ⊥ Y(x) | L.
Thomas Richardson Therme Vals Workshop Slide 44
X Y L H
(b-i)
X x Y(x) L H
(b-ii) Here we can read directly from the template that
X ⊥ ⊥ Y(x) | L.
Thomas Richardson Therme Vals Workshop Slide 45
A B C D H
A and C are treatments; H is unobserved; B is a time varying confounder; D is the final response; Treatment C is assigned randomly conditional on the observed history, A and B; Want to know P(D( ˜ a, ˜ c)).
Thomas Richardson Therme Vals Workshop Slide 46
A B C D H
If the following holds: A ⊥ ⊥ D( ˜ a, ˜ c) C( ˜ a) ⊥ ⊥ D( ˜ a, ˜ c) | B( ˜ a), A General result of Robins (1986) then implies: P(D( ˜ a, ˜ c)=d) =
P(B=b | A= ˜ a)P(D=d | A= ˜ a, B=b, C= ˜ c). Does it??
Thomas Richardson Therme Vals Workshop Slide 47
A
˜
a B( ˜ a) C( ˜ a)
˜
c D( ˜ a, ˜ c) H
d-separation:
A ⊥ ⊥ D( ˜ a, ˜ c) C( ˜ a) ⊥ ⊥ D( ˜ a, ˜ c) | B( ˜ a), A
General result of Robins (1986) then implies:
P(D( ˜ a, ˜ c)=d) =
P(B=b | A= ˜ a)P(D=d | A= ˜ a, B=b, C= ˜ c).
Thomas Richardson Therme Vals Workshop Slide 48
A B C D H UH UB UC UD a B(a) C(a) D(a) H(a) a B(a, c) c D(a, c) H(a, c)
Thomas Richardson Therme Vals Workshop Slide 49
A B C D H2 H1 A ⊥ ⊥ D( ˜ a, ˜ c) C( ˜ a) ⊥ ⊥ D( ˜ a, ˜ c) | B( ˜ a), A
General result of Robins (1986) then implies:
P(D( ˜ a, ˜ c)=d) =
P(B=b | A= ˜ a)P(D=d | A= ˜ a, B=b, C= ˜ c).
Does it??
Thomas Richardson Therme Vals Workshop Slide 50
A B C D H2 H1 A
˜
a B( ˜ a) C( ˜ a)
˜
c D( ˜ a, ˜ c) H1 H2 A ⊥ ⊥ D( ˜ a, ˜ c) C( ˜ a) ⊥ ⊥ D( ˜ a, ˜ c) | B( ˜ a), A
General result of Robins (1986) then implies:
P(D( ˜ a, ˜ c)=d) =
P(B=b | A= ˜ a)P(D=d | A= ˜ a, B=b, C= ˜ c).
Thomas Richardson Therme Vals Workshop Slide 51
Observed data:
O ≡ L1, A1, . . . , LK, AK, Y.
If the following holds for k = 1, . . . , K
Y(a†) ⊥ ⊥ Ak(a†) | Lk(a†), Ak−1(a†);
(7) then (under positivity):
P(Y(a†)=y | Lj(a†) = lj, Aj−1(a†) = a†
j−1)
=
p(y|lK, a†
K) K
p(lj|lj−1, a†
j−1).
(8)
Here Aj−1(a†) ≡ A1, . . . , Aj−1(a†
j−2), similarly for Lj−1(a†).
The RHS of (8) is referred to as the ‘g-formula’.
Thomas Richardson Therme Vals Workshop Slide 52
A dynamic regime g is a policy that assigns treatment (usually at multiple time points) on the basis of past history; Including conditional on the ‘natural’ value of treatment in the absence of an intervention; Exercise for as long as you would have done without intervention or twenty minutes, whichever is more. See Young et al. (2012) for additional analysis.
Thomas Richardson Therme Vals Workshop Slide 53
A1 a1 L(a1) A2(a1) a2 Y(a1, a2) H2 H1 A1
A+
1 (g)
L(g) A2(g)
A+
2 (g)
Y(g) H2 H1 P(Y(g)) is identified.
Thomas Richardson Therme Vals Workshop Slide 54
A1 a1 L(a1) A2(a1) a2 Y(a1, a2) H2 H1 A1
A+
1 (g)
L(g) A2(g)
A+
2 (g)
Y(g) H2 H1 P(Y(g)) is not identified.
Thomas Richardson Therme Vals Workshop Slide 55
We saw earlier that the causal DAG X → Y implied:
X ⊥ ⊥ Y(x0)
and
X ⊥ ⊥ Y(x1)
However, joint independence relations such as:
X ⊥ ⊥ Y(x0), Y(x1)
never follow from our SWIG transformation: There is no way via node-splitting to construct a graph with both
Y(x0), and Y(x1).
This has important consequences for the identification of direct effects.
Thomas Richardson Therme Vals Workshop Slide 56
Thomas Richardson Therme Vals Workshop Slide 57
Intervention on X and M:
M Y X M(˜ x)
˜
m X
˜
x Y(˜ x, ˜ m)
d-separation gives:
X ⊥ ⊥ M(˜ x) ⊥ ⊥ Y(˜ x, ˜ m)
Pearl associates additional independence relations with this graph
Y(x1, m) ⊥ ⊥ M(x0), X Y(x0, m) ⊥ ⊥ M(x1), X
equivalent to assuming independent errors, εX ⊥
⊥ εM ⊥ ⊥ εY.
Thomas Richardson Therme Vals Workshop Slide 58
Pure (aka Natural) Direct Effect (PDE): Change in Y had X been different, but M fixed at the value it would have taken had X not been changed:
PDE ≡ Y(x1, M(x0)) − Y(x0, M(x0)).
Legal motivation [from Pearl (2000)]: “The central question in any employment-discrimination case is whether the employer would have taken the same action had the employee been
else had been the same.” (Carson versus Bethlehem Steel Corp., 70 FEP Cases 921, 7th Cir. (1996)).
Thomas Richardson Therme Vals Workshop Slide 59
PDE also allows non-parametric decomposition of Total Effect (ACE) into direct (PDE) and indirect (TIE) pieces.
PDE ≡ E [Y(1, M(0))] − E [Y(0)] TIE ≡ E [Y (1, M(1)) − Y (1, M(0))] TIE + PDE ≡ E [Y(1)] − E [Y(0)] ≡ ACE(X → Y)
Thomas Richardson Therme Vals Workshop Slide 60
Pearl and others claim that under “no confounding” the PDE is identified by the following mediation formula: PDEmed(m) =
[E[Y|x1, m] − E[Y|x0, m]] P(m|x0)
Thomas Richardson Therme Vals Workshop Slide 61
Observational data on three variables:
X- treatment: cigarette cessation M intermediate: blood pressure at 1 year, high or low Y outcome: say CHD by 2 years
Observed data (X, M, Y) on each of n subjects. All binary
X randomly assigned
Thomas Richardson Therme Vals Workshop Slide 62
Y = 0 Y = 1
Total ˆ
P(Y =1 | m, x) M = 0
1500 500 2000 0.25
X = 0 M = 1
1200 800 2000 0.40
M = 0
948 252 1200 0.21
X = 1 M = 1
1568 1232 2800 0.44 A researcher, Prof H wishes to apply the mediation formula to estimate the PDE. Prof H believes that there is no confounding, so that Pearl’s NPSEM-IE holds, but his post-doc, Dr L is skeptical.
Thomas Richardson Therme Vals Workshop Slide 63
To try to address Dr L ’s concerns, Prof H carries out animal intervention studies.
Y = 0 Y = 1
Total ˆ
P(Y(m, x)=1) M = 0
750 250 1000 0.25
X = 0 M = 1
600 400 1000 0.40
M = 0
790 210 1000 0.21
X = 1 M = 1
560 440 1000 0.44 As we see: ˆ
P(Y(m, x)=1) = ˆ P(Y =1 | m, x);
Prof H is now convinced: ‘What other experiment could I do ?’ He applies the mediation formula, yielding
PDE
med = 0. Conclusion: No direct effect of X on Y.
Thomas Richardson Therme Vals Workshop Slide 64
Under the true generating process, the true value of the PDE is:
PDE
med = 0 Prof H’s conclusion was completely wrong!
Thomas Richardson Therme Vals Workshop Slide 65
Dr L was right – there was a confounder:
M Y X H
Y ⊥ ⊥ H | M, X = 0
and
M ⊥ ⊥ H | X = 1
Thomas Richardson Therme Vals Workshop Slide 66
Dr L was right – there was a confounder:
M Y X H
Y ⊥ ⊥ H | M, X = 0
and
M ⊥ ⊥ H | X = 1 M Y H X = 0 M Y H X = 1
The confounding undetectable by any intervention on X and/or M. Pearl: Onus is on the researcher to be sure there is no confounding. Causation should precede intervention.
Thomas Richardson Therme Vals Workshop Slide 67
If our only interventions are on the variables X and M then we cannot do an experiment to learn the PDE. We could learn E [Y{x = 1, M(x = 0)}] by intervention if we could
◮ intervene and set X to 0 and observe M(0), ◮ then return each subject to their pre-intervention state, ◮ finally intervene to set X to 1 and M to M(0) and observe
Y(1, M(0)). Such an intervention strategy will usually not exist because not possible in a real-world intervention (e.g., suppose the outcome Y were death). Because we cannot observe the same subject under both X = 1 and X = 0 (i.e. ”across worlds”,) no intervention will allow us to learn the distribution of mixed counterfactuals such as Y{x = 1, M(x = 0)} : (In the story Dr L had to introduce a new node on the graph in order to check the value of the PDE via an experiment.)
Thomas Richardson Therme Vals Workshop Slide 68
The independent error assumption cannot be checked by any randomized experiment on the variables in the graph. ⇒ Connection between experimental interventions and potential
⇒ Theories in Social and Medical sciences are not detailed enough to support the independent error assumption. What about faithfulness and causal discovery procedures? Such inferences are explicit that they rely on faithfulness, and are designed to guide hypothesis formation;
◮ Contrast: In Pearl’s NPSEM-IE approach the simple act of
using a DAG is viewed as automatically committing you to making this untestable hypothesis. Predictions (possibly derived assuming faithfulness) regarding intervention distributions P(Y(x)) = P(Y | do(x)) can be tested by randomized experiments.
Thomas Richardson Therme Vals Workshop Slide 69
Assumption of independent errors implies super-exponentially many ‘cross-world’ counterfactual independence assumptions:
2 3 4 K
3 7 15 2K − 1
3 7 15 2K − 1
7 127 32767 2(2K−1) − 1
5 113 32697 (2(2K−1) − 1) − K−1
j=1 (4j − 2j)
4 19 274 K−1
j=0 (22j − 1)
1 94 32423 O(22K−2) constrnts in NPSEM-IE Table: Dimensions of counterfactual models associated with complete graphs with binary variables.
Thomas Richardson Therme Vals Workshop Slide 70
In an NPSEM we define a counterfactual independence to be logical if it holds regardless of the distribution over counterfactuals (equivalently error terms) e.g. for binary X
Y(x0) ⊥ ⊥ Y(x1) | X, Y
Completeness Conjecture There exists a distribution over counterfactuals that is experimentally indistinguishable from the NPSEM that assumes independent errors but in which the
derived from the SWIG.
Thomas Richardson Therme Vals Workshop Slide 71
A simple approach to unifying graphs and counterfactuals via node-splitting The approach works via linking the factorizations associated with the two graphs The approach provides a language that allows counterfactual and graphical people to communicate The approach leads to many fewer untestable independence assumptions than in the NPSEM-IE approach of Pearl. The approach also provides a way to combine information on the absence of individual and population level direct effects.
Thomas Richardson Therme Vals Workshop Slide 72
Thomas Richardson Therme Vals Workshop Slide 73
Pearl, J. Causality (Second ed.). Cambridge, UK: Cambridge University Press, 2009. Richardson, TS, Robins, JM. Single World Intervention Graphs. CSSS Technical Report No. 128 http://www.csss.washington.edu/Papers/wp128.pdf, 2013. Robins, JM A new approach to causal inference in mortality studies with sustained exposure periods applications to control of the healthy worker survivor effect. Mathematical Modeling 7, 1393–1512, 1986. Robins, JM, VanderWeele, TJ, Richardson TS. Discussion of “Causal effects in the presence of non compliance a latent variable interpretation by Forcina, A. Metron LXIV (3), 288–298, 2007. Shpitser, I, Pearl, J. What counterfactuals can be tested. Journal of Machine Learning Research 9, 1941–1979, 2008. Spirtes, P , Glymour, C, Scheines R. Causation, Prediction and
Thomas Richardson Therme Vals Workshop Slide 74
Pearl correctly states that using his Twin Network method (next slide) it may be shown that Y(x0, x1) is not independent of X1, given Z and X0. However, he then goes on to say (incorrectly): In the twin network model there is a d-connected path from X1 to Y(x0, x1). . . Therefore, [(3)] is not satisfied for Y(x0, x1) and X1. [Ex. 11.3.3, p.353] This is actually incorrect in two ways: Y(x0, x1) ⊥ ⊥ X1 | Z, X0 does not imply Y(x0, x1) ⊥ ⊥ X1 | Z, X0 =x0 d-separation is not complete for Twin Networks so the presence of a d-connected path does not imply that an independence is not implied.
Thomas Richardson Therme Vals Workshop Slide 75
T X Z Y T X Z Y UT UX UZ UY T(z) X(z) z Y(z) T X Z z Y(z)
T and Y(z) are d-connected given X in the twin-network, but in spite of this T ⊥ ⊥ Y(z) | X under the associated NPSEM-IE because X(z) = X, and T and Y(z) are d-separated given X in the twin-network.
Thomas Richardson Therme Vals Workshop Slide 76
Intervention on Z alone.
X Y Z X(˜ z) Z
˜
z Y(˜ z)
factorization:
P(Z, X(˜ z), Y(˜ z)) = P(Z)P(X(˜ z))P(Y(˜ z) | X(˜ z))
modularity:
P(X(˜ z)=x) = P(X=x | Z= ˜ z), P(Y(˜ z)=y | X(˜ z)=x) = P(Y =y | X=x, Z= ˜ z).
d-separation gives:
Z ⊥ ⊥ X(˜ z), Y(˜ z).
Thomas Richardson Therme Vals Workshop Slide 77
Intervention on Z and X:
X Y Z X(˜ z)
˜
x Z
˜
z Y(˜ x, ˜ z)
factorization:
P(Z, X(˜ z), Y(˜ x, ˜ z)) = P(Z)P(X(˜ z))P(Y(˜ x, ˜ z))
modularity:
P(X(˜ z)=x) = P(X=x | Z= ˜ z), P(Y(˜ x, ˜ z)=y) = P(Y =y | X= ˜ x, Z= ˜ z).
d-separation gives:
Z ⊥ ⊥ X(˜ z) ⊥ ⊥ Y(˜ x, ˜ z)
Thomas Richardson Therme Vals Workshop Slide 78
Compare:
X Y Z X(˜ z) Z
˜
z Y(˜ z)
P(Z, X(˜ z), Y(˜ z)) = P(Z)P(X(˜ z))P(Y(˜ z) | X(˜ z)) P(Y(˜ z)=y | X(˜ z) = x) = P(Y =y | X=x, Z= ˜ z).
versus
X Y Z X(˜ z) Z
˜
z Y(˜ z)
P(Z, X(˜ z), Y(˜ z)) = P(Z)P(X(˜ z))P(Y(˜ z) | X(˜ z)), P(Y(˜ z)=y | X(˜ z) = x) = P(Y =y | X=x)
Thomas Richardson Therme Vals Workshop Slide 79
X Y Z X(˜ z) Z Y(˜ z)
P(Z, X(˜ z), Y(˜ z)) = P(Z)P(X(˜ z))P(Y(˜ z) | X(˜ z)) P(Y(˜ z)=y | X(˜ z) = x) = P(Y =y | X=x, Z= ˜ z).
versus
X Y Z X(˜ z) Z Y(˜ z)
P(Z, X(˜ z), Y(˜ z)) = P(Z)P(X(˜ z))P(Y(˜ z) | X(˜ z)), P(Y(˜ z)=y | X(˜ z) = x) = P(Y =y | X=x)
Red nodes are needed in order to read off modularity property from G(˜ a).
Thomas Richardson Therme Vals Workshop Slide 80
X Y Z X(˜ z) Z
˜
z Y(˜ z)
factorization:
P(Z, X(˜ z), Y(˜ z)) = P(Z)P(X(˜ z))P(Y(˜ z) | X(˜ z))
modularity:
P(X(˜ z)=x) = P(X=x | Z= ˜ z), P(Y(˜ z)=y | X(˜ z) = x) = P(Y =y | X=x).
d-separation gives:
Z ⊥ ⊥ X(˜ z), Y(˜ z)
Thomas Richardson Therme Vals Workshop Slide 81